SOLUTION: Multiply and simplify by factoring. Assume that all expressions under radicals represent non-negative numbers. cubic root of y^7 times cubic root of 81y^8=??? I am not sure hot

Algebra ->  Square-cubic-other-roots -> SOLUTION: Multiply and simplify by factoring. Assume that all expressions under radicals represent non-negative numbers. cubic root of y^7 times cubic root of 81y^8=??? I am not sure hot       Log On


   

Question 397070: Multiply and simplify by factoring. Assume that all expressions under radicals represent non-negative numbers.
cubic root of y^7 times cubic root of 81y^8=???
I am not sure hot to write this problem in the formula plotting system so if you could let me know how that would be great.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
You did a pretty good job of using words to describe your expression. Using Algebra.com's formula rendering feature, the expression
root%283%2C+y%5E7%29%2Aroot%283%2C+81y%5E8%29
is specified as:
root(3, y^7)*root(3, 81y^8)
surrounded by 3 left braces and 3 right braces. The "3" represents the type of root. So any kind of root can be done with this syntax.

With complicated radical expressions one would often simplify the two radicals before multiplying them. But your expression is not that complicated so we will go ahead and use a property of radicals, root%28a%2C+p%29%2Aroot%28a%2C+q%29+=+root%28a%2C+p%2Aq%29, to multiply the two radicals. This gives us:
root%283%2C+y%5E7%2A81y%5E8%29
Multiplying the y factors together we get:
root%283%2C+81y%5E15%29
Now we simplify the radical. Since this is a cube root we look for factors of the radicand (the expression inside a radical), that are perfect cubes. With a little effort we should find that 27 is a perfect cube, 3%5E3, and it is a factor of 81. And y%5E15 is a perfect cube itself since its exponent is a multiple of 3 (%28y%5E5%29%5E3+=+y%5E15). Factoring out the perfect cubes we get:
root%283%2C+27%2A3%2Ay%5E15%29
And using the Commutative property to change the order so the perfect cubes are in front. (This is an optional but recommended step.):
root%283%2C+27%2Ay%5E15%2A3%29
Then we use the same property of radicals we used earlier. Only this time we use it in the opposite direction (right to left) to split this one radical into the product of the radicals of each of its factors:
root%283%2C+27%29%2Aroot%283%2C+y%5E15%29%2Aroot%283%2C+3%29
The cube roots of the perfect cubes simplify:
3%2Ay%5E5%2Aroot%283%2C+3%29
which is the simplified form of your original expression.

Note: By using the Commutative property earlier the simplified cube roots of the perfect cubes end up in front of the remaining radical. This is the preferred way to write such an expression.